A Comparative Study of Classical and Modern Methods for Solving Boundary Value Problems

Abstract

Boundary value problems (BVPs) constitute a fundamental class of mathematical models describing physical and engineering systems governed by differential equations and prescribed boundary conditions. This study presents a comparative analytical-numerical framework for evaluating classical and modern BVP solvers, emphasizing numerical accuracy, convergence behavior, computational efficiency, stability, and applicability. We examine classical approaches, including separation of variables, eigenfunction expansions, Green's functions, and shooting methods, alongside finite difference, finite element, finite volume, spectral collocation, and iterative techniques. We conduct a computational experiment on a smooth one-dimensional Dirichlet boundary value problem with a known exact solution to evaluate finite difference, linear finite element, spectral collocation, and shooting methods. The results demonstrate that the finite difference method achieves an observed convergence order of 1.998, while the finite element prototype exhibits an observed nodal convergence order of 3.998 for the selected benchmark. At 128 degrees of freedom, the maximum numerical errors decrease to 4.942 × 10⁻⁵ and 2.443 × 10⁻¹⁰ for the finite difference and finite element methods, respectively. Spectral collocation achieves near machine-precision accuracy with relatively few degrees of freedom, while the shooting method recovers the exact initial slope with an absolute error of 1.499 × 10⁻¹² after two iterations. These findings demonstrate that numerical performance depends on solution regularity, discretization strategy, and problem structure. The study establishes a comparative framework that integrates analytical validation, numerical approximation, and statistical convergence assessment, providing a systematic basis for selecting appropriate solution methods for boundary value problems in applied mathematics and engineering.